
Construction of Fourier bases on directed graphs.
We propose the random walk operator associated with a random walk on a directed graph as a reference operator. We propose the eigenvectors of the random walk operator as a Fourier basis on directed graphs and determine a variational characterization of the eigenvectors of the random walk operator – see Proposition REF .

Construction of multi-resolution analyses on directed graphs.
We propose an overcomplete spectral wavelet transform on directed graphs in spectralwaveletdg. This construction extends the framework of spectral wavelets on undirected graphs [1]} and diffusion polynomial frames [2]}. We also propose a critically sampled wavelet transform in Section REF  generalizing the framework of diffusion wavelets [3]}, [4]} to the directed case.

Efficiency of the theoretical framework through applications.
The development of our harmonic analysis on directed graphs leads us to consider semi-supervised learning problems with regularization of type \(\ell _2\)  in Section REF  and \(\ell _1\)  regularization in section REF  and signal modeling by filtering on directed graphs in Section REF  highlighting the effectiveness of our theoretical framework.

